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Search: a111035 -id:a111035
Displaying 1-10 of 10 results found. page 1
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A331977 Integers m such that m and m+1 are terms of A111035. +20
2
1, 6479, 11663, 34943, 47519, 51983, 196559, 327359, 685583, 954239, 1016063, 1346879, 2039183, 2332799, 2504447, 4665599, 5143823, 5962319, 6128639, 6723359, 7225343, 9363599, 12027023, 12446783, 14930351, 17639999, 17735759, 22924943, 24681023, 34715519, 41990399 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
Daniel Suteu, Table of n, a(n) for n = 1..128
Daniel Yaqubi and Amirali Fatehizadeh, Some results on average of Fibonacci and Lucas sequences, arXiv:2001.11839 [math.CO], 2020.
PROG
(PARI) f(n, m) = (Mod([1, 1; 1, 0], m)^n)[1, 2];
isok(n) = f(n+2, n)==1 && f(n+3, n+1)==1;
for(k=1, 10^7, if(isok(k), print1(k, ", "))); \\ Daniel Suteu, Feb 03 2020
CROSSREFS
Cf. A111035 (the sum of the first k nonzero Fibonacci numbers is divisible by k).
KEYWORD
nonn
AUTHOR
Michel Marcus, Feb 03 2020
EXTENSIONS
a(23)-a(31) from Daniel Suteu, Feb 03 2020.
STATUS
approved
A101907 Numbers n-1 such that the arithmetic mean of the first n Fibonacci numbers (beginning with F(0)) is an integer. +10
6
0, 3, 5, 8, 10, 18, 23, 28, 30, 33, 40, 45, 47, 58, 60, 70, 71, 78, 88, 93, 95, 99, 100, 105, 108, 119, 128, 130, 138, 143, 148, 150, 165, 178, 180, 190, 191, 198, 200, 210, 213, 215, 219, 225, 228, 238, 239, 240, 248, 250, 268, 270, 273, 280, 287, 310, 320, 330 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
The sum of the first n Fibonacci numbers is F(n+2)-1, sequence A000071.
Knott discusses the factorization of these numbers. - T. D. Noe, Oct 10 2005
LINKS
T. D. Noe, Table of n, a(n) for n = 1..1000
Y. Bugeaud, F. Luca, M. Mignotte and S. Siksek, On Fibonacci numbers with few prime divisors, Proc. Japan Acad., 81, Ser. A (2005), pp. 17-20.
H. R. Morton, Fibonacci-like sequences and greatest common divisors, The American Mathematical Monthly, Vol. 102, No. 8 (October 1995), pp. 731-734.
M. Ward, The prime divisors of Fibonacci numbers, Pacific J. Math., Vol. 11, No. 1 (1961), pp. 379-386.
Eric W. Weisstein's World of Mathematics, Arithmetic mean
Eric W. Weisstein's World of Mathematics, Fibonacci
FORMULA
Numbers n-1 such that (F(0)+ F(1)+ ... + F(n-1)) / n is an integer. F(i) is the i-th Fibonacci number.
a(n) = A219612(n) - 1. - Altug Alkan, Dec 29 2015
EXAMPLE
n=4 : (F(0)+F(1)+F(2)+F(3))/4 = (0+1+1+2)/4 = 1. So n-1 = 4-1 = 3 is a term.
n=6 : (F(0)+F(1)+F(2)+F(3)+F(4)+F(5))/6 = (0+1+1+2+3+5)/6 = 2. So n-1 = 6-1 = 5 is a term.
MATHEMATICA
Select[ Range[0, 500], Mod[Fibonacci[ # + 2] - 1, # + 1] == 0 &] (* Robert G. Wilson v *)
PROG
(PARI) is(n)=((Mod([1, 1; 1, 0], n+1))^(n+2))[1, 2]==1 \\ Charles R Greathouse IV, Feb 04 2013
CROSSREFS
Cf. A000045, A000071. See A111035 for another version.
Cf. A219612. - Altug Alkan, Dec 29 2015
KEYWORD
easy,nonn
AUTHOR
Ctibor O. Zizka, Jul 27 2008
EXTENSIONS
Edited and extended by Robert G. Wilson v, Aug 03 2008
Definition corrected by Altug Alkan, Dec 29 2015
STATUS
approved
A124456 Numbers n which divide the sum of the Fibonacci numbers F(1) through F(n) and such that n is not a multiple of 24. +10
6
1, 2, 77, 319, 323, 1517, 3021, 4757, 6479, 7221, 8159, 8229, 9797, 11663, 12597, 13629, 13869, 14429, 14949, 16637, 18407, 19043, 19437, 23407, 24947, 25437, 30049, 30621, 34943, 34989, 35207, 39203, 43677, 44099, 47519, 51983, 53663, 55221, 65471, 70221, 77837, 78089, 79547 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Numbers n which divide the sum of the first n nonzero Fibonacci numbers are listed in A111035 = {1, 2, 24, 48, 72, 77, 96, ...}. Most of these are multiples of 24. These multiples divided by 24 are listed in A124455 = {1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, ...}. [Edited by M. F. Hasler, Feb 04 2020]
A111035(2024) = 758642 is in this sequence but not in A331976. - Don Reble, Feb 04 2020
The even terms a({2, 155, 397, 469, ...}) = {2, 758642, 7057466, 10805846, ...} are now listed in A331870. - M. F. Hasler, Feb 06 2020
LINKS
Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 200 terms from M. F. Hasler)
FORMULA
{ n != 0 (mod 24) | A000071(n+2) == 0 (mod n) }. - M. F. Hasler, Feb 06 2020
MATHEMATICA
Select[Range[20000], !IntegerQ[ #/24]&&Mod[Fibonacci[ #+2]-1, # ]==0&]
PROG
(PARI) A124456_vec(N=44, n=0)={vector(N, i, until( n++%24&& is_A111035(n), ); n)} \\ M. F. Hasler, Feb 04 2020
(Sage) [n for n in (1..20000) if mod(n, 24)!=0 and mod(fibonacci(n+2)-1, n)==0 ] # G. C. Greubel, Feb 16 2020
CROSSREFS
Cf. A000045, A111035, A124455.
Cf. A331976 (odd terms).
KEYWORD
nonn
AUTHOR
Alexander Adamchuk, Nov 02 2006, Nov 03 2006
EXTENSIONS
Edited by M. F. Hasler, Feb 04 2020
STATUS
approved
A331976 Odd integers m that divide the sum of the first m nonzero Fibonacci numbers. +10
4
1, 77, 319, 323, 1517, 3021, 4757, 6479, 7221, 8159, 8229, 9797, 11663, 12597, 13629, 13869, 14429, 14949, 16637, 18407, 19043, 19437, 23407, 24947, 25437, 30049, 30621, 34943, 34989, 35207, 39203, 43677, 44099, 47519, 51983, 53663, 55221, 65471, 70221, 77837, 78089, 79547 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Not all terms are squarefree, for instance 13869, 14949, 43677, 93357, ... are not.
A subsequence of A124456, missing just the even terms A124456({2, 155, 397, 469, ...}) = {2, 758624, 7057466, 10805846, ...}. - M. F. Hasler, Feb 06 2020
LINKS
Giovanni Resta, Table of n, a(n) for n = 1..10000
Daniel Yaqubi and Amirali Fatehizadeh, Some results on average of Fibonacci and Lucas sequences, arXiv:2001.11839 [math.CO], 2020.
FORMULA
Equals A124456 without even terms. - M. F. Hasler, Feb 06 2020
PROG
(PARI) forstep(n=1, oo, 2, is_A111035(n) && print1(n", ")) \\ M. F. Hasler, Feb 06 2020
CROSSREFS
Odd terms in A111035, A124456.
KEYWORD
nonn
AUTHOR
Michel Marcus, Feb 03 2020
STATUS
approved
A111058 Numbers k such that the average of the first k Lucas numbers is an integer. +10
3
1, 2, 8, 12, 20, 24, 48, 60, 68, 72, 92, 96, 120, 140, 144, 188, 192, 200, 212, 216, 240, 288, 300, 332, 336, 360, 384, 428, 432, 440, 452, 480, 500, 548, 576, 600, 648, 660, 668, 672, 680, 692, 696, 720, 768, 780, 788, 812, 864, 908, 932, 960, 1008, 1028, 1052 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
A111035 is the equivalent for Fibonacci numbers and has many elements in common with this sequence. T. D. Noe, who extended this sequence, noticed that, for some reason, 24 divides many of those k.
All terms are even except for the first term. - Harvey P. Dale, Apr 22 2024
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from Harvey P. Dale)
FORMULA
k such that (Sum_{i=1..k} A000204(i))/k is an integer.
{ k : A027961(k) == 0 (mod k) }. - Alois P. Heinz, Apr 23 2024
MATHEMATICA
Lucas[n_] := Fibonacci[n+1]+Fibonacci[n-1]; lst={}; s=0; Do[s=s+Lucas[n]; If[Mod[s, n]==0, AppendTo[lst, n]], {n, 1000}]; lst (* T. D. Noe *)
Module[{nn=1000, ln}, ln=LucasL[Range[nn]]; Table[If[IntegerQ[Mean[Take[ln, n]]], n, Nothing], {n, nn}]] (* Harvey P. Dale, Apr 22 2024 *)
CROSSREFS
Cf. A000032, A000204, A027961, A111035, A160665.
KEYWORD
easy,nonn
AUTHOR
Jonathan Vos Post, Oct 07 2005
STATUS
approved
A331870 Even numbers n which divide the sum of the Fibonacci numbers F(1) + ... + F(n) but are not a multiple of 24. +10
3
2, 758642, 7057466, 10805846, 50860946, 59677526, 61800878, 155045678, 178551374, 217281146, 343943882, 359455694, 432175586, 609069506, 1449599486, 1721358698, 1829675354, 1884592706, 2013264194, 2116706282, 2680549946, 2971193186, 3084402122, 3252387386, 3454785386 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
A111035 lists numbers n which divide the sum of the first n nonzero Fibonacci numbers. Most of these are multiples of 24. Sequence A124456 lists those which aren't. Most of these are odd (cf. A331976), this sequence lists the exceptions.
a(2) was found by Don Reble, cf. A124456.
If we consider F(n+2) = 1 + the sum of the first n nonzero Fibonacci numbers (cf. A000071), then for even n we find:
4 divides F(n+2) for n == 4 (mod 12), 3 divides F(n+2) for n == 6 (mod 12),
F(n+2) == 3 (mod 4) for n == 8 (mod 12), 2 divides F(n+2) for n == 10 (mod 12),
F(n+2) == 5 (mod 6) for n == 12 (mod 24).
These relations imply that all terms a(n) == 2 (mod 12) for all n. This also means that all terms of A111035 are either divisible by 24, or odd, or congruent to 2 (mod 12).
LINKS
Giovanni Resta, Table of n, a(n) for n = 1..80
FORMULA
a(n) == 2 (mod 12) for all n.
PROG
(PARI) M=[1, 1; 1, 0]; forstep(n=2, oo, 12, n%24&&(Mod(M, n)^(n+1))[1, 1]==1&& print1(n", ")) \\ Custom implementation of is_A111035(), check for updates there.
CROSSREFS
Cf. A124456, A331976, A111035, A000045 (Fibonacci numbers), A000071 (F(n)-1 = F(0)+...+F(n-2)).
KEYWORD
nonn
AUTHOR
M. F. Hasler, Feb 29 2020
EXTENSIONS
Terms a(15) and beyond from Giovanni Resta, Mar 02 2020
STATUS
approved
A124455 Numbers n such that 24n divides the sum of the first 24n nonzero Fibonacci numbers. +10
2
1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 24, 25, 27, 28, 30, 32, 36, 40, 42, 45, 46, 48, 50, 51, 54, 55, 56, 57, 60, 64, 70, 72, 75, 80, 81, 84, 86, 90, 92, 96, 98, 100, 102, 108, 110, 112, 114, 120, 125, 126, 128, 135, 138, 140, 144, 150, 153, 155, 160, 162 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Numbers n such that n divides the sum of the first n nonzero Fibonacci numbers are listed in A111035 = {1, 2, 24, 48, 72, 77, 96, 120, 144, 192, 216, 240, 288, 319, 323, 336, 360, ...}. Most of these are multiples of 24. Those which are not a multiple of 24 are listed in A124456 = {1, 2, 77, 319, 323, 1517, 3021, 4757, 6479, 7221, 8159, 8229, 9797, ...}.
This sequence coincides with A072378 (12n | F(12n)) for all values up to 84. The first two different terms are 86 and 164.
Prime a(n) are {2, 3, 5, 281, ...}.
LINKS
Table of n, a(n) for n=1..65.
MATHEMATICA
Select[Range[10000], IntegerQ[ #/24]&&Mod[Fibonacci[ #+2]-1, # ]==0&] /24
CROSSREFS
Cf. A111035, A124456, A072378 (numbers n such that 12n divides Fibonacci(12n)), A000045 (Fibonacci numbers).
KEYWORD
nonn
AUTHOR
Alexander Adamchuk, Nov 02 2006
EXTENSIONS
Edited by M. F. Hasler, Feb 04 2020
STATUS
approved
A160757 Averages of the Fibonacci numbers which take integer values. +10
1
1, 1, 5058, 262213938, 18124577012898, 187952389930860, 1409394295257361938, 116903055445824294157698, 10100618828005365858877129458, 81435914480042681825934186407384633298, 7505278652741640947693896415563573183203138, 700346071081054203480884565881868806176873272498 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
COMMENTS
The numbers n such that F(1)+F(2)+...+F(n)=F(n+2)-1 is divisible by n are given in A111035. [From Max Alekseyev, Jun 04 2009]
LINKS
Table of n, a(n) for n=1..12.
FORMULA
1/n*Sum {j=1..n} Fibonacci_j is an integer.
a(n) = (A000045(A111035(n)+2)-1) / A111035(n) = A000071(A111035(n)+2) / A111035(n) [From Max Alekseyev, Jun 04 2009]
MATHEMATICA
lst = {}; Do[a = Sum[ Fibonacci@ j, {j, n}]/n; If[ IntegerQ@ a, AppendTo[lst, a]], {n, 250}]; lst
CROSSREFS
Cf. A050248, integer average of n primes for some n, A000045.
KEYWORD
easy,nonn
AUTHOR
Daniel Tisdale, May 25 2009
EXTENSIONS
Corrected and extended by Max Alekseyev and Robert G. Wilson v, Jun 04 2009
STATUS
approved
A282772 Starting from F(n), minimum number, greater than 1, of consecutive Fibonacci numbers whose average is an integer. +10
1
4, 2, 3, 12, 2, 13, 3, 2, 6, 5, 2, 12, 4, 2, 3, 12, 2, 24, 3, 2, 6, 24, 2, 12, 4, 2, 3, 12, 2, 5, 3, 2, 6, 13, 2, 12, 4, 2, 3, 5, 2, 24, 3, 2, 5, 24, 2, 12, 4, 2, 3, 12, 2, 24, 3, 2, 6, 24, 2, 5, 4, 2, 3, 12, 2, 24, 3, 2, 6, 5, 2, 12, 4, 2, 3, 12, 2, 24, 3, 2, 6 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
Entries are 2, 3, 4, 5, 6, 12, 13 and 24.
Periodic with period equal to 420.
LINKS
Paolo P. Lava, Table of n, a(n) for n = 0..1000
FORMULA
a(3*k + 1) = 2;
a(12*k + 2) = a(12*k + 6) = 3;
a(12*k) = 4;
a(30*k + 9) = a(30*k + 29) = a(60*k + 44) = 5;
a(60*k + 8) = a(60*k + 20) = a(60*k + 32) = a(60*k + 56) = 6;
a(60*k + 3) = a(60*k + 11) = a(60*k + 15) = a(60*k + 23) = a(60*k + 27) = a(60*k + 35) = a(60*k + 47) = a(60*k + 51) = 12;
a(420*k + 5) = a(420*k + 33) = a(420*k + 117) = a(420*k + 173) = a(420*k + 201) = a(420*k + 257) = a(420*k + 285) = a(420*k + 341) = 13;
a(420*k + x) = 24, with x = 17, 21, 41, 45, 53, 57, 65, 77, 81, 93, 101, 105, 113, 125, 137, 141, 153, 161, 165, 177, 185, 197, 213, 221, 225, 233, 237, 245, 261, 273, 281, 293, 297, 305, 317, 321, 333, 345, 353, 365, 377, 381, 393, 401, 405, 413, 417.
EXAMPLE
a(0) = 4 because F(0) + F(1) + F(2) + F(3) = 0 + 1 + 1 + 2 = 4 and 4/4 = 1;
a(1) = 2 because F(1) + F(2) = 1 + 1 = 2 and 2/2 = 1;
a(2) = 3 because F(2) + F(3) + F(4) = 1 + 2 + 3 = 6 and 6/3 = 2;
a(3) = 12 because F(3) + F(4) + ... + F(13) + F(14) = 2 + 3 + ... + 233 + 377 = 984 and 984/12 = 82.
MAPLE
with(combinat): P:=proc(q) local a, k, n; for k from 0 to q do a:=fibonacci(k); for n from 1 to q do a:=a+fibonacci(k+n);
if type(a/(n+1), integer) then print(n+1); break; fi; od; od; end: P(10^3);
MATHEMATICA
Table[k = 1; While[! IntegerQ@ Mean@ Take[#, n ;; n + k], k++]; k + 1, {n, Length@ # - 24}] &@ Fibonacci@ Range[0, 419] (* Michael De Vlieger, Mar 06 2017 *)
CROSSREFS
Cf. A000045, A101907, A111035, A254141.
KEYWORD
nonn,easy
AUTHOR
Paolo P. Lava, Mar 03 2017
STATUS
approved
A339598 Numbers k such that k | Sum_{i=1..k} A000045(i) and k+1 | Sum_{i=1..k+1} A000045(i). +10
0
1, 6479, 11663, 51983, 196559 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
a(1)-a(5) are given on p. 11 in Yaqubi, Fatehizadeh, 2020. According to the authors there are no other terms up to 10^6.
Apparently an erroneous version of A331977.
Included in accordance with OEIS policy of including published but erroneous sequences to serve as pointers to the correct values.
LINKS
Table of n, a(n) for n=1..5.
D. Yaqubi and A. Fatehizadeh, Some results on average of Fibonacci and Lucas sequences, arXiv:2001.11839 [math.CO], 2020.
PROG
(PARI) is(n) = my(s=sum(i=1, n, fibonacci(i))); lift(Mod(s, n))==0 && lift(Mod(s+fibonacci(n+1), n+1))==0
CROSSREFS
Cf. A000045, A111035.
KEYWORD
nonn,hard,more
AUTHOR
Felix Fröhlich, Dec 09 2020
STATUS
approved
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